Causes of the apparent scale independence of fractal indices associated with forest fragmentation in Bolivia

In a recent investigation, Landsat TM and ETM+ data were used to simulate different resolutions of remotely-sensed images (from 30 to 1100 m) and to analyze the effect of resolution on a range of landscape metrics associated with spatial patterns of forest fragmentation in Chapare, Bolivia since the mid-1980s. Whereas most metrics were found to be highly dependent on pixel size, several fractal metrics (DLFD, MPFD, and AWMPFD) were apparently independent of image resolution, in contradiction with a sizeable body of literature indicating that fractal dimensions of natural objects depend strongly on image characteristics. The present re-analysis of the Chapare images, using two alternative algorithms routinely used for the evaluation of fractal dimensions, shows that the values of the box-counting and information fractal dimensions are systematically larger, sometimes by as much as 85%, than the "fractal" indices DLFD, MPFD, and AWMFD for the same images. In addition, the geometrical fractal features of the forest and non-forest patches in the Chapare region strongly depend on the resolution of images used in the analysis. The largest dependency on resolution occurs for the box-counting fractal dimension in the case of the non-forest patches in 1993, where the difference between the 30 and I 100 m-resolution images corresponds to 24% of the full theoretical range (1.0 to 2.0) of the mass fractal dimension. The observation that the indices DLFD, MPFD, and AWMPFD, unlike the classical fractal dimensions, appear relatively unaffected by resolution in the case of the Chapare images seems due essentially to the fact that these indices are based on a heuristic, "non-geometric" approach to fractals. Because of their lack of a foundation in fractal geometry, nothing guarantees that these indices will be resolution-independent in general. (C) 2006 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.

Exploring the spatial variation of take-all (Gaeumannomyces graminis var. tritici) for site-specific management

The fungus Gaeumannomyces graminis var. tritici (Ggt), commonly known as the take-all fungus, causes damage to roots of wheat and barley that limits crop growth and causes loss of yield. There was little knowledge on the within-field spatial variation of take-all and relations with features in the growing crop, selected soil properties and spectral information from remotely sensed imagery. Geostatistical analyses showed that take-all, chlorosis and leaf area index had similar patchy distributions. Many of the spectral bands from a hyperspectral image also had similar spatial patterns to take-all and chlorosis. Relations between take-all and mineral nitrogen, elevation and pH were generally weaker.

Using the variogram to explore imagery of two different spatial resolutions

The resolution of remotely sensed data is becoming increasingly fine, and there are now many sources of data with a pixel size of 1 m x 1 m. This produces huge amounts of data that have to be stored, processed and transmitted. For environmental applications this resolution possibly provides far more data than are needed: data overload. This poses the question: how much is too much? We have explored two resolutions of data-20 in pixel SPOT data and I in pixel Computerized Airborne Multispectral Imaging System (CAMIS) data from Fort A. P. Hill (Virginia, USA), using the variogram of geostatistics. For both we used the normalized difference vegetation index (NDVI). Three scales of spatial variation were identified in both the SPOT and 1 in data: there was some overlap at the intermediate spatial scales of about 150 in and of 500 m-600 in. We subsampled the I in data and scales of variation of about 30 in and of 300 in were identified consistently until the separation between pixel centroids was 15 in (or 1 in 225pixels). At this stage, spatial scales of about 100m and 600m were described, which suggested that only now was there a real difference in the amount of spatial information available from an environmental perspective. These latter were similar spatial scales to those identified from the SPOT image. We have also analysed I in CAMIS data from Fort Story (Virginia, USA) for comparison and the outcome is similar.:From these analyses it seems that a pixel size of 20m is adequate for many environmental applications, and that if more detail is required the higher resolution data could be sub-sampled to a 10m separation between pixel centroids without any serious loss of information. This reduces significantly the amount of data that needs to be stored, transmitted and analysed and has important implications for data compression.

Estimating the spatial scale of herbicide and soil interactions by nested sampling, hierarchical analysis of variance and residual maximum likelihood

An unbalanced nested sampling design was used to investigate the spatial scale of soil and herbicide interactions at the field scale. A hierarchical analysis of variance based on residual maximum likelihood (REML) was used to analyse the data and provide a first estimate of the variogram. Soil samples were taken at 108 locations at a range of separating distances in a 9 ha field to explore small and medium scale spatial variation. Soil organic matter content, pH, particle size distribution, microbial biomass and the degradation and sorption of the herbicide, isoproturon, were determined for each soil sample. A large proportion of the spatial variation in isoproturon degradation and sorption occurred at sampling intervals less than 60 m, however, the sampling design did not resolve the variation present at scales greater than this. A sampling interval of 20-25 m should ensure that the main spatial structures are identified for isoproturon degradation rate and sorption without too great a loss of information in this field.

Spatial analysis of the error in a model of soil nitrogen

Models of the dynamics of nitrogen in soil (soil-N) can be used to aid the fertilizer management of a crop. The predictions of soil-N models can be validated by comparison with observed data. Validation generally involves calculating non-spatial statistics of the observations and predictions, such as their means, their mean squared-difference, and their correlation. However, when the model predictions are spatially distributed across a landscape the model requires validation with spatial statistics. There are three reasons for this: (i) the model may be more or less successful at reproducing the variance of the observations at different spatial scales; (ii) the correlation of the predictions with the observations may be different at different spatial scales; (iii) the spatial pattern of model error may be informative. In this study we used a model, parameterized with spatially variable input information about the soil, to predict the mineral-N content of soil in an arable field, and compared the results with observed data. We validated the performance of the N model spatially with a linear mixed model of the observations and model predictions, estimated by residual maximum likelihood. This novel approach allowed us to describe the joint variation of the observations and predictions as: (i) independent random variation that occurred at a fine spatial scale; (ii) correlated random variation that occurred at a coarse spatial scale; (iii) systematic variation associated with a spatial trend. The linear mixed model revealed that, in general, the performance of the N model changed depending on the spatial scale of interest. At the scales associated with random variation, the N model underestimated the variance of the observations, and the predictions were correlated poorly with the observations. At the scale of the trend, the predictions and observations shared a common surface. The spatial pattern of the error of the N model suggested that the observations were affected by the local soil condition, but this was not accounted for by the N model. In summary, the N model would be well-suited to field-scale management of soil nitrogen, but suited poorly to management at finer spatial scales. This information was not apparent with a non-spatial validation. (c),2007 Elsevier B.V. All rights reserved.

Water quality and ecology of the River Lee: mass balance and a review of temporal and spatial data

A regional overview of the water quality and ecology of the River Lee catchment is presented. Specifically, data describing the chemical, microbiological and macrobiological water quality and fisheries communities have been analysed, based on a division into river, sewage treatment works, fish-farm, lake and industrial samples. Nutrient enrichment and the highest concentrations of metals and micro-organics were found in the urbanised, lower reaches of the Lee and in the Lee Navigation. Average annual concentrations of metals were generally within environmental quality standards although, oil many occasions, concentrations of cadmium, copper, lead, mercury and zinc were in excess of the standards. Various organic substances (used as herbicides, fungicides, insecticides, chlorination by-products and industrial solvents) were widely detected in the Lee system. Concentrations of ten micro-organic substances were observed in excess of their environmental quality standards, though not in terms of annual averages. Sewage treatment works were the principal point source input of nutrients. metals and micro-organic determinands to the catchment. Diffuse nitrogen sources contributed approximately 60% and 27% of the in-stream load in the upper and lower Lee respectively, whereas approximately 60% and 20% of the in-stream phosphorus load was derived from diffuse sources in the upper and lower Lee. For metals, the most significant source was the urban runoff from North London. In reaches less affected by effluent discharges, diffuse runoff from urban and agricultural areas dominated trends. Flig-h microbiological content, observed in the River Lee particularly in urbanised reaches, was far in excess of the EC Bathing Water Directive standards. Water quality issues and degraded habitat in the lower reaches of the Lee have led to impoverished aquatic fauna but, within the mid-catchment reaches and upper agricultural tributaries, less nutrient enrichment and channel alteration has permitted more diverse aquatic fauna.

Estimating the spatial scales of regionalized variables by nested sampling, hierarchical analysis of variance and residual maximum likelihood

The variogram is essential for local estimation and mapping of any variable by kriging. The variogram itself must usually be estimated from sample data. The sampling density is a compromise between precision and cost, but it must be sufficiently dense to encompass the principal spatial sources of variance. A nested, multi-stage, sampling with separating distances increasing in geometric progression from stage to stage will do that. The data may then be analyzed by a hierarchical analysis of variance to estimate the components of variance for every stage, and hence lag. By accumulating the components starting from the shortest lag one obtains a rough variogram for modest effort. For balanced designs the analysis of variance is optimal; for unbalanced ones, however, these estimators are not necessarily the best, and the analysis by residual maximum likelihood (REML) will usually be preferable. The paper summarizes the underlying theory and illustrates its application with data from three surveys, one in which the design had four stages and was balanced and two implemented with unbalanced designs to economize when there were more stages. A Fortran program is available for the analysis of variance, and code for the REML analysis is listed in the paper. (c) 2005 Elsevier Ltd. All rights reserved.

A singular vector perspective of 4DVAR: The spatial structure and evolution of baroclinic weather systems

The extent to which the four-dimensional variational data assimilation (4DVAR) is able to use information about the time evolution of the atmosphere to infer the vertical spatial structure of baroclinic weather systems is investigated. The singular value decomposition (SVD) of the 4DVAR observability matrix is introduced as a novel technique to examine the spatial structure of analysis increments. Specific results are illustrated using 4DVAR analyses and SVD within an idealized 2D Eady model setting. Three different aspects are investigated. The first aspect considers correcting errors that result in normal-mode growth or decay. The results show that 4DVAR performs well at correcting growing errors but not decaying errors. Although it is possible for 4DVAR to correct decaying errors, the assimilation of observations can be detrimental to a forecast because 4DVAR is likely to add growing errors instead of correcting decaying errors. The second aspect shows that the singular values of the observability matrix are a useful tool to identify the optimal spatial and temporal locations for the observations. The results show that the ability to extract the time-evolution information can be maximized by placing the observations far apart in time. The third aspect considers correcting errors that result in nonmodal rapid growth. 4DVAR is able to use the model dynamics to infer some of the vertical structure. However, the specification of the case-dependent background error variances plays a crucial role.